Transient-Time Fractional-Space Trigonometry and Application
In this work, we use the generalized exponential function in the fractional-order domain to define generalized cosine and sine functions. We then re-visit some important trigonometric identities and generalize them from the narrow integer-order subset to the more general fractional-order domain. It is clearly shown that trigonometric functions and trigonometric identities in the transient-time of a non-integer-order system have significantly different values from their steady-state values. Identities such as sin2(t) + cos2(t) = 1 are shown to be invalid in the transient-time of a fractional-order system. Some generalized hyperbolic functions and identities are also given in this work. Application to the evaluation of the step-response of a non-integer-order system is given.
KeywordsFractional-Calculus Generalized Exponential function Fractional order systems Generalized Trigonometric Functions Generalized Hyperbolic Functions
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